Intersection theorems on polytopes

Intersection theorems are used to prove the existence of solutions to mathematical programming and game theoretic problems. The known intersection theorems on the unit simplex are the theorems of Knaster–Kuratowski–Mazurkiewicz (KKM), Scarf, Shapely, Freund, and Ichiishi. Recently the intersection result of KKM has been generalized by Ichiishi and Idzik to closed coverings of a compact convex polyhedron, called a polytope. In this paper we formulate a general intersection theorem on the polytope. To do so, we need to generalize the concept of balancedness as is used by Shapley and by Ichiishi. The theorem implies most of the results stated above as special cases. First, we show that the theorems of KKM, Scarf, Shapley, Freund, and Ichiishi on the unit simplex and also some theorems of Ichiishi and Idzik on a polytope all satisfy the conditions of our theorem on the polytope. Secondly, the general theorem allows us to formulate the analogs of these theorems on the polytope.